28.4.34 problem 7.34

Internal problem ID [4566]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.34
Date solved : Monday, January 27, 2025 at 09:23:34 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 65

dsolve([diff(x__1(t),t)=3*x__1(t)-x__2(t)+x__3(t),diff(x__2(t),t)=x__1(t)+x__2(t)+x__3(t),diff(x__3(t),t)=4*x__1(t)-x__2(t)+4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{5 t}+c_{2} {\mathrm e}^{t}+{\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{5 t}+c_{2} {\mathrm e}^{t}-2 \,{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= 3 c_{1} {\mathrm e}^{5 t}-c_{2} {\mathrm e}^{t}-3 \,{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 169

DSolve[{D[x1[t],t]==3*x1[t]-x2[t]+x3[t],D[x2[t],t]==x1[t]+x2[t]+x3[t],D[x3[t],t]==4*x1[t]-x2[t]+4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{12} e^t \left (c_1 \left (4 e^t+5 e^{4 t}+3\right )-2 c_2 \left (2 e^t+e^{4 t}-3\right )+3 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{12} e^t \left (c_1 \left (-8 e^t+5 e^{4 t}+3\right )+c_2 \left (8 e^t-2 e^{4 t}+6\right )+3 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{4} e^t \left (c_1 \left (-4 e^t+5 e^{4 t}-1\right )-2 c_2 \left (-2 e^t+e^{4 t}+1\right )+c_3 \left (3 e^{4 t}+1\right )\right ) \\ \end{align*}